Riemannian Geometry 9781400884216

Table of contents :
Contents
CHAPTER I Tensor analysis
1. Transformation of coördinates. The summation convention
2. Contravariant vectors. Congruences of curves
3. Invariants. Covariant vectors
4. Tensors. Symmetric and skew-symmetric tensors
5. Addition, subtraction and multiplication of tensors. Contraction
6. Conjugate symmetric tensors of the second order. Associate tensors
7. The Christoffel 3-index symbols and their relations
8. Riemann symbols and the Riemaun tensor. The Ricci tensor
9. Quadratic differential forms
10. The equivalence of symmetric quadratic differential forms
11. Covariant differentiation with respect to a tensor gij
CHAPTER II Introduction of a metric
12. Definition of a metric. The fundamental tensor
13. Angle of two vectors. Orthogonality
14. Differential parameters. The normals to a hypersurface
15. N-tuply orthogonal systems of hypersurfaces in a Vn
16. Metric properties of a space Vn immersed in a Vm
17. Geodesics
18. Riemannian, normal and geodesic coördinates
19. Geodesic form of the linear element. Finite equations of geodesics
20. Curvature of a curve
21. Parallelism
22. Parallel displacement and the Riemann tensor
23. Fields of parallel vectors
24. Associate directions. Parallelism in a sub-space
25. Curvature of Vn at a point
26. The Bianchi identity. The theorem of Schur
27. Isometric correspondence of spaces of constant curvature. Motions in a Vn
28. Conformal spaces. Spaces conformal to a flat space
CHAPTER III Orthogonal ennuples
29. Determination of tensors by means of the components of an orthogonal ennuple and invariants
30. Coefficients of rotation. Geodesic congruences
31. Determinants and matrices
32. The orthogonal ennuple of Schmidt. Associate directions of higher orders. The Frenet formulas for a curve in a Vn
33. Principal directions determined by a symmetric covariant tensor of the second order
34. Geometrical interpretation of the Ricci tensor. The Ricci principal directions
35. Condition that a congruence of an orthogonal ennuple be normal
36. N-tuply orthogonal systems of hypersurfaces
37. N-tuply orthogonal systems of hypersurfaces in a space conformal to a flat space
38. Congruences canonical with respect to a given congruence
39. Spaces for which the equations of geodesics admit a first integral
40. Spaces with corresponding geodesics
41. Certain spaces with corresponding geodesics
CHAPTER IV The geometry of sub-spaces
42. The normals to a space Vn immersed in a space Vm
43. The Gauss and Codazzi equations for a hypersurface
44. Curvature of a curve in a hypersurface
45. Principal normal curvatures of a hypersurface and lines of curvature
46. Properties of the second fundamental form. Conjugate directions. Asymptotic directions
47. Equations of Gauss and Codazzi for a Vn immersed in a Vm
48. Normal and relative curvatures of a curve in a Vn immersed in a Vm
49. The second fundamental form of a Vn in a Vm. Conjugate and asymptotic directions
50. Lines of curvature and mean curvature
51. The fundamental equations of a Vn in a Vm in terms of invariants and an orthogonal ennuple
52. Minimal varieties
53. Hypersurfaces with indeterminate lines of curvature
54. Totally geodesic varieties in a space
CHAPTER V Sub-spaces of a flat space
55. The class of a space Vn
56. A space Vn of class p > 1
57. Evolutes of a Vn in an Sn+p
58. A subspace Vn of a Vm immersed in an Sm+p
59. Spaces Vn of class one
60. Applicability of hypersurfaces of a flat space
61. Spaces of constant curvature which are hypersurfaces of a flat space
62. Coördinates of Weierstrass. Motion in a space of constant curvature
63. Equations of geodesics in a space of constant curvature in terms of coördinates of Weierstrass
64. Equations of a space Vn immersed in a Vm of constant curvature
65. Spaces Vn conformal to an Sn
CHAPTER VI Groups of motions
66. Properties of continuous groups
67. Transitive and intransitive groups. Invariant varieties
68. Infinitesimal transformations which preserve geodesics
69. Infinitesimal conformal transformations
70. Infinitesimal motions. The equations of Killing
71. Conditions of integrability of the equations of Killing. Spaces of constant curvature
72. Infinitesimal translations
73. Geometrical properties of the paths of a motion
74. Spaces V2 which admit a group of motions
75. Intransitive groups of motions
76. Spaces V2 admitting a G2 of motions. Complete groups of motions of order n(n+1)/2 – 1
77. Simply transitive groups as groups of motions
Bibliography
Index

Citation preview

RIEMANNIAN GEOMETRY

PR IN C ETO N LANDMARKS IN M A T H E M A T IC S AND P H Y S IC S

Non-standard Analysis, by Abraham Robinson General Theory of Relativity, by R A M Dirac Angular Momentum in Quantum Mechanics, by A. R. Edmonds Mathematical Foundations of Quantum Mechanics, by John von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar

Riemannian Geometry, by Luther Pfahler Eisenhart

RIEMANNIAN GEOMETRY BY

LU TH ER PFAHLER EISEN H ART

PRINCETON UNIVERSITY PRESS PRINCETON,

NEW JER SEY

P ublished by Princeton University Press, 41 W illiam Street, P rinceton, New Jersey 08540 In the U n ited Kingdom: Princeton University Press, C hichester, West Sussex All Rights Reserved ISBN 0-691-08026-7 P rinceton University Press books are printed on acid-free paper and m eet the gu id elines for p erm an en ce and durability o f the C om m ittee on Production G uidelines for Book Longevity o f the C ouncil on Library Resources S econ d Printing, 1950 Third Printing, 1952 Fourth Printing, 1960 Fifth Printing, 1964 Seventh Printing, 1993 Eighth Printing, for the P rinceton Landmarks in M athem atics and Physics series, 1997 Printed in the U n ited States o f Am erica 9

11

13

15

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12

10 8

Preface The recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann pro­ posed the generalization, to spaces of any order, of the theory of surfaces, as developed by Gauss, and introduced certain fundamental ideas in this general theory. Prom time to time important con­ tributions to the theory were made by Bianchi, Beltrami, Christoffel, Schur, Voss and others, and Ricci coordinated and extended the theory with the use of tensor analysis and his Absolute Calculus. Recently there has been an extensive study and development of Riemannian Geometry, and this book aims to present the existing theory. Throughout the book constant use is made of the methods of tensor analysis and the Absolute Calculus of Ricci and Levi-Civita. The first chapter contains an exposition of tensor analysis in form and extent sufficient for the reader of the book who has not previously studied this subject. However, it is not intended that the exposition shall give an exhaustive foundational treatment of the subject. Most, if not all, of the contributors to the theory of Riemannian Geometry have limited their investigations to spaces with a metric defined by a positive definite quadratic differential form. How­ ever, the theory of relativity deals with spaces with an indefinite fundamental form. Consequently the former restriction is not made in this book. Although many results of the older theory have been modified accordingly, much remains to be done in this field. The theory of parallelism of vectors in a general Riemannian manifold, as introduced by Levi-Civita and developed by others, is set forth in the second chapter and is applied in other parts of the book. The extensions of this theory to non-Riemannian geometries are not developed in this book, since it is my intention to present some of them in a later book.

Preface

vi

Of the many exercises in the book some involve merely direct applications of the formulas of the text, but most of them con­ stitute extensions of the theory which might properly be included as portions of a more extensive treatise. References to the sources of these exercises are given for the benefit of the reader. All references in the book are to the papers listed in the Bibliography. In the writing of this book I have had invaluable assistance and criticism by four of my students, Dr. Arthur Bramley, Dr. Harry Levy, Dr. J. H. Taylor and Dr. J. M. Thomas. I desire also to express my appreciation of the courtesies extended by the printers Liitcke & Wulff and by the Princeton University Press. October, 1925.

Luther Pfahler Eisenhart.

Contents Ch a p t e r

I

section Tensor analysis Page 1 . Transformation of coordinates. The summation convention................. 1 2. Contravariant vectors. Congruences of curves...................................... 3 6 3. Invariants. Covariant vectors........................................... 4. Tensors. Symmetric and skew-symmetric tensors................................ 9 5. Addition, subtraction and multiplication of tensors. Contraction 12 6. Conjugate symmetric tensors of the second order. Associate tensors . 14 7. The Christoffel 3-index symbols and their relations.............................. 17 S. Riemann symbols and the Riemaun tensor. The Ricci te n so r 19 9. Quadratic differential forms...................................................................... 22 10. The equivalence of symmetric quadratic differential forms................... 23 11 . Covariant differentiation with respect to a tensor gtJ.......................... 26 Ch a p t e r

II

Introduction of a metric 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Definition of a metric. The fundamental te n so r.............. Angle of two vectors. Orthogonality..................................................... Differential parameters. The normals to a hypersurface.................... N-tu$\y orthogonal systems of hypersurfaces in a Fn.......................... Metric properties of a space F» immersed in a 7m ........................ Geodesics................................... Riemannian, normal and geodesic coordinates . ..................................... Geodesic form of the linear element. Finite equations of geodesics .. Curvature of a cu rv e................................................................................ Parallelism..................................................................................... Parallel displacement and the Riemann tensor..................................... Fields of parallel vectors.......................................................................... Associate directions. Parallelism in a sub-space................................. Curvature of Fn at a poin t...................................................................... The Bianchi identity. The theorem of Schur ................ Isometric correspondence of spaces of constant curvature. Motions in a Vn............................ 28. Conformal spaces. Spaces conformal to a flat sp a ce .......................... Ch a p t e r

34 37 41 43 44 48 53 57 60 62 65 67 72 79 82 84 89

III

Orthogonal ennuples 29.

Determination of tensors by means of the components of anorthogonal ennuple and invariants...............................................................................

96

viii

Contents

Section Page 30. Coefficients of rotation. Geodesic congruences................................. 97 31. Determinants and matrices................................................................... 101 32. The orthogonal ennuple of Schmidt. Associate directions of higher orders. The Frenet formulas for a curve in a Fn............................ 103 33. Principal directions determined by a symmetric covariant tensor ofthe second o rd er........................................................................................... 107 34. Geometrical interpretation of the Ricci tensor.The Ricci principal directions............................ 113 35. Condition that a congruence of an orthogonal ennuple be normal-114 36. jV-tuply orthogonal systems of hypersurfaces.................................... 117 37. JV-tuply orthogonal systems of hypersurfaces in a space conformal to ....... 119 a flat space 38. Congruences canonical with respect to a given congruence............ 125 39. Spaces for which the equations of geodesics admit a first integral. . . 128 40. Spaces with corresponding geodesics.................................................. 131 41. Certain spaces with corresponding geodesics..................................... 135 Ch a p t e b

IV

The geometry of sub-spaces 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.

The normals to a space F» immersed in a space F«............................. 143 The Gauss and Codazzi equations for a hypersurface....................... . 146 Curvature of a curve in a hypersurface............................................ 150 Principal normal curvatures of a hypersurface and lines of curvature. 152 Properties of the second fundamental form. Conjugate directions. Asymptotic directions................................ 155 Equations of Gauss and Codazzi for a Fn immersed in a F « ........ 159 Normal and relative curvatures of a curve in a Fn immersed in a F« 164 The second fundamental form of a Fn in a Vm. Conjugate and asymp­ totic directions................................................... 106 Lines of curvature and mean curvature.................................................. 167 The fundamental equations of a F« in a F™ in terms of invariantsand an orthogonal ennuple........................................................................... 170 Minimal varieties.................................................................................. 176 Hypersurfaces with indeterminate lines of curvature............................ 179 Totally geodesic varieties in a sp a c e..................................................... 183 Ch a p t e b

V

Sub-spaces of a fiat space 55. The class of aspace Fn

56. 57. 58. 59. 60.

.................................................................. 187 A space Fw of class j? > 1 ......................................................................... 189 Evolutes of a Vn in an Sn+P .................................................................... 192 A subspace Fn of a F« immersed in an ..................... 195 Spaces Fn of class one............................................................................... 197 Applicability of hypersurfaces of a flat space........................................... 200

Contents Section

ix Page

61. 62. 63.

Spaces of constant curvature which are hypersurfaces of aflat space 20 L Coordinates of Weierstrass. Motion in a space of constantcurvature 204 Equations of geodesics in a space of constant curvature in terms of coordinates of Weierstrass.....................................................................207 64. Equations of a space F« immersed in a Vm of constant curvature-210 65. Spaces Vn conformal to an Sn............................................................. 214 C h a p te r VI

Groups of motions 66. 67. 68. 69. 70. 71.

72. 73. 74. 75. 76. 77.

Properties of continuous groups......................................................... 221 Transitive and intransitive groups. Invariant varieties.......... ........ 225 Infinitesimal transformations which preserve geodesics................... 227 Infinitesimal conformal transformations...............................................230 Infinitesimal motions. The equations of Killing..............................233 Conditions of integrability of the equations of Killing. Spaces of constant curvature..................................................................................237 Infinitesimal translations....................................................................... 239 Geometrical properties of the paths of a motion............................. 240 Spaces Vs which admit a group of motions....................................... 241 Intransitive groups of m otions........................................................... 244 Spaces F* admitting a 6r* of motions. Complete groups ofmotions of order n (n + 1)/2 — 1.............................................................................. 245 Simply transitive groups as groups of motions................................. 247 Bibliography.......................................................................................... 252

CHAPTER I

Tensor analysis i.■,°- ), T ra n s fo rm a tio n o f c o o rd in a te s. T h e s u m m a tio n co n ­ ven tio n . Any n independent variables x i, where i takes the values 1 to n, may be thought of as the coordinates of an n -dimensional space Y„ in the sense that each set of values of the variables defines a point of P». Unless stated otherwise it is understood that the coordinates are real. Suppose that we have n independent real functions a ,,...,'.- a ,,

9 * Sl

9x*m

a covariant tensor o f the mth order; ( 4 71

a,r' "•r-

=

d x 'r '

...

8 x ''m d x h daf m d x 'Pi

...

d x 'Pq

a mixed tensot' o f them -\-q order which is contravariant o f the mth order• and covariant o f the qth order* Concerning these definitions we make the following observations and deductions: (1) A superscript indicates contravariant character, a subscript covariant; (2) Any set of functions in sufficient number can be taken as the components of a tensor of any type and order in one coordinate system and the components in any other system are defined by equations (4.5), (4.6) or (4.7) as the case may be; (3) A contravariant vector is a contravariant tensor of the first order; a covariant vector is a covariant tensor of the first order; (4) An invariant is a tensor of zero order. The latter designation is a more appropriate term than invariant because of the possible ambiguity of the term invariant; (5) From (4.5), (4.6) and (4.7) it follows that if the components of a tensor in one coordinate system are zero at a point, they are * It can be shown as in § 2 that these definitions possess the group property.

4. Tensors. Symmetric and skew-symmetric tensors

11

zero at this point in every coordinate system; in particular, if the components are identically zero in one coordinate system, they are identically zero in every coordinate system. From the form of equations (4.5), (4.6) and (4.7) it is clear that the order ofthe indices plays a role in these equations.Suppose, however', that the relative position in the a’s of two or more indices, either contravariant or covariant, is immaterial, which means that the a ’s with these indices interchanged are equal. Then from the form of these equations it follows that the order of the corre­ sponding indices in the a"s is immaterial. For example, suppose that in (4.5) aSlS*'"Sm = as-Sl'" Sn, then we have a ’r i - - r™ —

_

a firi a *i*»•••»»S x

M-

a /r0 a 8x . . . dx

d x $l

dx'1

SX

9X

cfjrji•. Multiplication and contraction may be combined to give tensors. Thus from the tensors ay and brst we may obtain a tensor of the third order, such as aij bjst, or aij bHt, or a vector as aij W*. This combined process is referred to by some writers as inner multi­ plication. We remark that this process was used in (3.4). Let aylm be a set of functions of ad and a'ffa be a set of functions of x ' 1 such that a)jlm l l and a'*fc l ' v are the components ofa tensor, when X1 is an arbitrary vector. From this hypothesis and in con­ sequence of (4.7) and (2.2) we have .,v

n i; — aklm

d x ’a d x ' t dad d x m doc1 dx 'a % ,y T T

OX

ox1

dxm *’ * .

ox

,*•

Since X v is arbitrary, we have ,«p ® Iiva

4.;

d x * d x f^ dzj

docP

dx1 dxm

d x ,„

d x ,r dx>c ,

and consequently a%m and a,aJ*va are the components of a mixed tensor of the fifth order. This proof applies equally well when any of the subscripts is used for contraction with I.1; also a similar result can be established if the arbitrary vector is covariant. Since * Ricci and Lcvi-Civita, 1901,1, p. 133 call the process composition, and German writers, Yerjiingnng.

14

I* Tensor analysis

the proof is not conditioned by the number of indices of the functions a, we have the following theorem of which the theorem of § 3 is a particular case: Given a set o f functions o f x* and a set a'st^ f stmo f x n , ij

,.p APl and

X.*1 are components o f a tensor in

the coordinates x * and x ' 1 respectively, when I* and Xn are com­ ponents o f an arbitrary vector in these respective coordinates, the given functions are components o f a tensor o f one higher order. A similar theorem holds if X* is replaced by a tensor of any type and one of the indices is contracted. This is sometimes called the quotient law of tensors. 6. C on ju g ate s y m m e tric te n so rs o f th e se c o n d o rd e r. A sso ciate te n so rs. Let gy be the components of a symmetric covariant tensor of the second order, that is, gg — gji. We denote by g the determinant of the g f s, that is, gn ■■• gm (6 .1)

g =

.

. . .

.

(Jnl * * * (jnn

If gij denotes the cofactor of (6-2)

divided by g , we have

giJg,cj =

Si,

where dl have the values (1.5). For it follows from the definition of gij that when i ^ k the left-hand member of (6 .2 ) is the sum of the product of the terms of one row (or column) of (6 .1) by the cofactors of another row (or column) divided by g\ and when i — k, this sum is equal to gig. Let X* be the components of an arbitrary vector, then gyX* is an arbitrary vector, say yj. Now by (6 .2 ) f J yj — g^gijX* = b\X* == Xk. Since yj is an arbitrary vector, we have as a consequence of the last theorem of § 5: I f 9 is the determinant o f a symmetric covariant tensor gij, the cofactors o f g^ divided by g and denoted by giJ are the components o f a symmetric contravariant tensor.

6. Conjugate symmetric tensors

15

It is clear that in like manner if giJ are the components of a symmetric contravariant tensor, the cofactors of gij in the determinant of the giJ,s divided by the determinant are the components of a symmetric covariant tensor of the second order. In either case we say that the tensor obtained by this process is the conjugate of the given one. As a consequence of the above result and (6 .2 ) we have that S\ are the components of a mixed tensor, which was proved directly in § 4. If in (6 .2 ) we replace k by i and sum for i, we get n terms each of which is unity. Hence for the invariant obtained from a symmetric tensor of the second order and its conjugate we have gvgy = n .

(6.3)

If we denote by ~g the determinant of giJ, we have by the rule for multiplying determinants and (6 .2 )

7

8

(I )

= j ^ T lik >® ~ { ik

« + [hi> ® •

Hence from (8.3), (8.5) and (8.7) we obtain (8 .8 ) Rhijk =

g ^ [ V .A ] + {

In consequence of (7.1) and (7.2) this is reducible to j} 8 91l\). [hj,

_

1 ( d*9hk

2 \da?dxi

|

d*9ij d*ghj d*gac \ da?da? da? da? da? dxi) + ([i j , m] [hk, 1}- [ik, m]

8. Biemann symbols and the Biemann tensor. The Bicci tensor

21

From (8.9) we find that the symbols of the first kind satisfy the following identities: (8.10) and (8.11)

Rhijk ==

Rthjk,

Rhijk ==:

Rhikj,

Rhijk ==

Rjkhi,

Rhijk + Rhjki + Rhkij =

0.

From (8.10) it follows that not more than two of the indices can be alike without the components vanishing; the same is true if the first two or second two indices are alike. Because of (8.10) there are n (n — 1 )/2 ( = ru) ways in which the first pair of indices are like the second pair, and w2 («2 — 1 )/2 ways in which the first pair and second pair are unlike; hence there is a total of m2 (w« + 1)/2 distinct symbols as regards (8.10). However, there are n ( n — 1 ) (n — 2 ) (n — 3)/4! ( = n4) equations of the form (8.11). Consequently there are (w2 -f l)/2 — « 4 = n*(n2— 1)/12 distinct symbols of the first kind.* In consequence of (8.10) we have from (8 .8 ) ( 8 .1 2 )

+

Jfev =

Also from (8.10) and (8.5) we have (8.13)

R liJk =

— R ilj k. t

If R lijk be contracted for I and k, we have, in consequence of (7.9), the tensor R ,j whose components are given by

(8.14)

r>.. = J

_ v

SH ogVg dx* dxJ

8 J k I \m 1 | k \ dxP 1i j ( \ i k ) \ m j i J __ Jm l jJlogVg_ \ijf dxm ’

* Cf., Christoffel, 1869, 1, p. 55. t Ricci and Levi Civita, 1901, 1, p. 142 denote R ^ as defined by (8.12) by aa kJ, and Bianchi, 1902, 1, p. 73 denotes it by (ih ,k j). Also the latter puts {it, kj} = g*h (ih, kj); hence {il,k j} is equal to — R \ %by (8.13).

22

I* Tensor analysis

which evidently is symmetric* We call the tensor Ry the Ricci tensor, as it was first considered by Ricci who gave it a geometrical interpretation in case gy is the fundamental tensor of a Riemann space (cf. §34).* Exercises 1. If R\jk in (8.3) is contracted for I and i, the resulting tensor is a zero tensor. 2. If K i j == e g y , then $ = --*■R, where B = gft B y . 3. Show from (7.14) that for transformations x '%= 9*(a1, • • •, xn~l), x ’n == x* the Christoffel symbols wher6 i, j 1, • ••, n — 1, are the componentsof a symmetric covariant tensor in a variety off* const.; likewise |^.J- and are the components of a mixed tensor and a covariant vector respectively. 4. Show that the tensor equation a{. A. = a k^ where a is an invariant, can be written in the form (a^ —

dh [h\ dof

Ij k f

30

I- Tensor analysis

Consequently we have A*,jk

(1 1 * 1 4 )

kj === Aj ffiijk j

where Rhjk is given by (8.3). In like manner for a tensor a%j we find (11.15) aij, ki -— a?y, ik = dih R hjki + O'hj R hm } and in general 1- • -m

(11.16)

Orr --rm,kl

Or,. • rm,»■

A necessary and sufficient condition that the Christoffel symbols be zero is that all of the gy s be constant, as follows from (7.1) and (7.4). Combining this result with the second theorem of § 10, we have the theorem: * Ricci and Levi+Civiia, 1901, 1, p. 143.

Exercises

31

I n order that there exist a coordinate system in which the first covariant derivatives with respect to a tensor gij reduce to ordinary derivatives at every point in space, it is necessary and sufficient that the Riemann symbols formed with respect to gij be zero and that the x ’s be those fo r which gij are constants. (Cf. § 18.) Exercises 1. The second theorem of § 11, and the identities (11.16) and (11.18) are con­ sequences of the definitions of covariant differentiation and do not involve an assumption that the quantities differentiated are components of tensors. 2. By applying the general rule of covariant (differentiation of § IX to the invariant k* p. show that this rule implies that the covariant derivative of an invariant is the ordinary derivative. 3. The tensor defined by a. • • *«r .

a . •.. a

&. •••O f

is called the contravariant derivative of respect to gy. Show that g*j>k = 0. Ricci and Levi-Civita, 1901, 1, p. 140. 4. If Oij is the curl of a covariant vector, show that

V'ij,k~b ajk, i

aid,j = 0,

and that this is equivalent to 3 Oij 3as* Is this condition sufficient as well as necessary that a skew-symmetric tensor a# 3aki CLjk . 3 he the curl of a vector? Eisenhart, 1922, 1. dx* 5. By definition a are the components of a relative tensor of weight p, if the equations connecting the components in two coordinate systems are of the form ,a p y a o£

j P ijk d x a d x ^ d x ’y dx1 dxm aim 9 ^ 9 ^ ' 9^; dx'^ d x 's ’ 9 yf where J is the J a c o b ia n — . Show that if Oij is a covariant tensor, then dx the cofactor of ay in the determinant |a#| is a relative contravariant tensor of weight two. 6. If aap is a covariant tensor of rank n — 1 (cf, Ex. 7, p. 16), there exist two relative vectors k* and , both of weight one, such that the cofactor of aap is of the form A*P = When aap is symmetric, ka and are the same relative vectors. 7. When a relative tensor is of weight one it is called a tensor density, Show that if the components of any tensor are multiplied by the square root of the non-vanishing determinant of a covariant tensor, they are the components of a tensor density.

32

I*Tensor analysis

8. The invariant is called the divergence of the vector 1* with respect to the symmetric tensor gy, Show that

lt,i = ~ y j 9. Show that the divergence of the tensor aij with respect to the symmetric tensor gy, that is, has the expression

and that the last term vanishes, if a'* is skew-symmetric. 10. The divergence of a mixed tensor a/ is reducible to

= y j Show that if the associate tensor a? is symmetric, 1

9

1

^ /„ ,•i f ~ \ * 1 „

1

2°

a‘-i ~~ y j dxi

da? ~ y j

9sci

dgpk

2 ***9ajf'

Einstein, 1916, 1, p. 799. 11. When gy and aij are the components of two symmetric tensors, if g*, au — gnajk + gjkOn — guaij = 0 ( i , j , k , l = 1, . . . . n), then dy = $gy. 12. If dijhi is a tensor satisfying the conditions (8.10) and for a vector I* we have I* Qyki = 0, a coordinate system x " can be chosen for which dyu are zero, when one or more of the indices is w. 13. Let for i = 1, • • •, n denote the components of n independent contra­ variant vectors, where the value of a for a = 1, **•, n indicates the vector (cf. Ex. 3, p. 8), and let A? denote the cofactor of la f in the determinant A = \ka f\ divided by A. Show that the quantities A** for each coordinate system are the components of a covariant vector, « indicating the vector and i the component. 14. Show that if a*#* 1 h \ h f h \k = 0 for any two arbitrary vectors and 12]*, then a?iijk -j- ctMji + ciyihk + djkxi = 0; also when dhyz possesses the properties (8.10) and (8.11), then a*#* = 0. 15. Show that when in a Vt the coordinates can be chosen (Cf. § 15) so that the components of a tensor gy are zero when i 4=j , then Rhj “ Rhk —

9« 9),

. • *, Xn)

(fc =JP +

1,

• • -,

the functions fe-, •• and fej •. rw for which n , • • •, rm take the values p + 1, • • •, n and in which we put (1) x i = x i = ai, where the a’s are constants, are components of the same tensor in the Vn P defined by (1). Levy, 1925, 1.

18. If gy and gy are the components of two symmetric tensors, and j Aj>and j /. are the corresponding Christoffel symbols, then by defined by

\ij\ = ,

.

and consequently |cos 0 | as defined by (13.2) is not greater than one. * When dealing with more than one vector, we usually make use of the notation and la\i to denote the contravariant and covariant components of one of several vectors, where the value of a indicates the vector and i the component. In the present case « takes the values 1 and 2.

38

H. Introduction of a, metric

When the components are not chosen so that the vectors be unit vectors, we have (13.3)

cos# =

.............. M h t M .... .... v (ftgy All* Air0 (e, gu X,\k A8]0

as follows from (12.6). If do? and da? denote differentials for two curves through a point, neither of which is a curve of length zero, we have (18.4)

cos» =

V(fixgy dx 1 dxJ) (es gid 6 xk 6 a})

,

When (12.3) is definite, a necessary and sufficient condition that two non-null vectors at a point be orthogonal is (13.5)

gy Ai|*Ajj|^ = 0,

and when the form is indefinite this is taken as the definition of orthogonality. The problem of determining vector-fields orthogonal to a given field will be treated later. When one, or both, of the given vectors is a null vector, the right-hand member of (13.2) involves an indeterminate factor, since there is no analogue to unit vectors in this case. Accordingly in retaining (13.2) as the definition of angle, this indeterminateness is understood. Furthermore, we take (13.5) as the definition of orthogonality when one or both of the vectors is null. As a consequence, a null vector is self-orthogonal. For the curves of parameter x i of the space we have dxl^Q, ■dxi = 0 , (j ^ i). Hence, when they are not minimal, the com­ ponents of the contravariant unit tangent vector are A* = 1IVcnga, AJ — 0 ( j 4 From this and (13.3) it follows that the angle

1 d lo g f f«

I i 1= l i t ’/

1 9logp» 2 9P

From (8.9) we have in this case Rhyk = 0 „

_ ,/•— ( S2V g u hiik — ^ W a p

2

dxj

’

(ft, i,j, k 4-),

d V g u d lo g V g hk 9P 1 9p

(16.S)

* . - vs

(F U •>£) + ^ _L 2 > J - i V j j L i ^ j m g-mm dx dx J

"£) (/i + i).

where ]£' indicates the sum for m — 1 , • • -, n excluding m = h m

and »i = i. 16. Metric properties of a space Vn immersed in a Vm. Consider a space Vm referred to coordinates f i and with the fundamental form (16.1) 9> — a«|j dya d t/.* If we put (16.2) f i = / “(P, •••,*»), * In this section Greek indices are supposed to take the values 1, • *•, m and Latin indices

16. Metric properties of a space Fn immersed in a F»

45

where the f i s are analytic functions of the x ’s such that the

3f “

matrix -~~r is of rank n, equations (16.2) define a space V„ ox 1 immersed in Vm. If we write (16,3)

= 9ih

then from the definition of linear element for Vm, namely ds 2 — eaup d y a difi,

(16.4)

we have for the linear element of Vn ds 2 =

(16.5)

egy d x * dxi.

Thus when a metric is defined for a space Ym, the metric of a sub­ space is in general determined (cf. Ex. 8 , p. 48). This is an evident generalization of the case of a surface a? = /* ( « , v) (for i — 1 , 2 , 3) in a euclidean space with the linear element (12.1); in this case (16.5) assumes the well-known form ds 2 = E d u 2 + 2 F d u dv + O d v 2 in the notation of Gauss. The formula for Vm analogous to (13.4) is (16.6)

aa8 d y a SyP cos 9 = — r - - v (ex a„p d ytt d i f ) (e2 aa/S 6 rja dy?)

From (16.2) we have (16.7)

d r = -^ d a ? .

Substituting in (16.6) and making use of (16.3), we obtain (13.4). Thus the invariant cos 9 of two directions at a point of V„ has the same value whether determined by the formula for F» or for the enveloping space Fm. Later (§ 55) it will be shown that when the fundamental form of a space is positive definite there exists a euclidean space Vm, where m < n ( n - f-l)/2 in which V„ can be considered as immersed. Consequently angle as defined by (13.4) for V„ is equal to the angle in the euclidean sense as determined in

46

II* Introduction of a metric

the enveloping Ym. In fact, in the differential geometry of a surface in euclidean 3-space, the angle between two directions on a surface is determined in the euclidean space and its expression in terms of the metric of the surface is derived therefrom; this gives a form of which (13.4) is an immediate generalization.* 'If I* are the components of any contravariant vector-field in V„, along any curve of the congruence of curves for which these are d oft dthe ytt tangent vectors we have - j j - — l l. From (16.7) we have for ~dt ~ this curve in Vm 9ya d x* Jx* d t ~ Hence the

9 ya . components in the y ’s of this vector-field are givenby dx* '

(16.8)

=

Conversely, if we have any vector-field £“ in Vm, for those vectors of the field in F„, that is, tangential to Vn, the components X* in V, the x ’s are obtained by taking anyWi of equations (16.8), gi} Vreplacing the y ’s by the expressions (16.2) and solving for the X’s. From (16.8) and (16.3) we have aafs F & =

(16.9)

and from (13.3) for two non-null vector-fields

COS0 =

(16.10)

■ :

..

X- ..... :■■■

..=

-

V (®1 aap ^1| ^11^) (e2a«p ^2|° ?2|^) . ; gq *i| Xj\J_________ Y (eigq X f V ) {e2gq X2f V )

3 va for a = 1, • • •, wi and a given i ox 1 are the components in the y’s of the tangents to the curves of From (16.7) it follows that

* Cf. Eisenhart, 1909, 1, p. 78. f n suitable equations.

Exercises

parameter xl in V„.

47

Since the matrix

dya

rank n by

hypothesis, there are n such independent vector-fields in Vn in terms of whose components the components of any vector-field in V„ are linearly expressible. From this it follows that any m functions I '3 satisfying the n equations aa?

(16.11) are the components such that the vector in Fn at the point. ponents & satisfying in the form

= 0

in the y 's of a vector-field at points of Vn, at a point of Vn is orthogonal to every vector Accordingly we say that a vector of com­ (16.11) is normal to Fn. If (16.11) is written =

(16.12)

we see that there are m — n linearly independent vector-fields normal to Vn. Exercises. 1. Show that a real coordinate system can be found for which g = 1 or — 1. In this coordinate system the divergence of a vector k* (Ex. 8, p. 32) is the ordinary divergence. 2. For a Fs referred to an orthogonal system of parametric curves

11 ^22

22 yl\

1221’

IS

’

2R

B = *B, = - f f - , ^11 ^22

and consequently R

TT (■£),-«•

In view of this result we have that if the constant in (17.9) is positive, negative or zero at a point of an integral curve of (17.8), it is the same all along the curve; that is, if the tangent vector at one point is non-null or null, the tangents all along the curve are of the same kind. From (17.7) it is seen that the form of (17.8) is not changed if s is replaced by as-j-5, where a and b are arbitrary constants. Hence, if the curve is not of length zero, s can be chosen so that (17.9) becomes (12.12), that is, s is the arc. On the other hand, if the constant in (17.9) is zero, the above mentioned generality of s obtains. Any integral curve of equations (17.8) is called a geodesic. When in particular it is a curve of length zero, we will call it a minimal geodesic, and we will understand that when s is used as a parameter of a minimal geodesic it is such that the differential equations of the geodesic assume the form (17.8). Consider for example the V4 of special relativity with the fundamental form

i

( d x ' l\

( d x '1 d x j \

,

*

where the a ’s are constants, we have: When the coordinates x* o f a space are subjected to an arbitrary analytic transformation, the Riemannian coordinates determined by the x’s and a point undergo a linear transformation with constant coefficients. dx'1 Since the a’s in (18.11) are the values of at the point, it is evident that conversely when a linear transformation of the Riemannian coordinates is given, corresponding analytic trans­ formations of the x’s exist but are not uniquely defined. At the point P 0 the coefficients ~gy in (18.4) are constants. From § 9 it follows that real linear transformations of the y's with constant coefficients can be found for which (18.4) reduces to a form at P 0 involving only squares of the differentials and the signs of these terms depend upon the signature of the differ­ ential form. These particular Riemannian coordinates have been See called normal coordinates by Birkhoff.* App.3 The transformation defined by (18.2) belongs to the class of transformations of the type * 1923, 2, p. 124.

56

H. Introduction of a metric

(18.12) x i — a.i + x 'l + Y C ttpx '“x ' } + -£YCapr x ' ax'Px'" ----where the c’s are symmetric in the subscripts. From (18.12) we have at P 0 of coordinates xo and x ,%= 0 in the resipective systems

Hence if {

indicates the Christoffel symbols in the x"s, we

have from (7.14)

{/*:I = Therefore a necessary and sufficient condition that that cjfc = — j

( i\ ’

— 0 is

. Accordingly the equations

x = x 0+ x ’ 1— y | (18.13)

x ,a x ’1*+ j j - caj3r x ,a x ’? x ,y -i----°

' .. .J

A

. . . x ' °m-4- • ■•

where the c’s are arbitrary constants symmetric in the subscripts,* define a transformation of coordinates such that (18-14)

( c) 0 = °\ dx Jo

The x ns so defined are called geodesic coordinates. Hence: At the origin o f a geodesic coordinate system first covariant derivatives are ordinary derivatives. The equations in geodesic coordinates of the geodesic through / dx^\ the origin determined by = r ^ r ) 3X6 (18.15)

x* = r * -

(r «^)o F & P s 3 ------

* This assumption is no restriction as to generality.

19.

Geodesic form of the linear element. Finite equations of geodesics

57

Comparing these expressions with (18.1) we see that Riemannian coordinates are the geodesic coordinates for which the T s vanish for P = 0 . 19 . G eo d esic fo rm o f th e lin e a r elem en t. F in ite eq u atio n s of g eo d esics. If f i x 1, ■• •, xn) is any real function such that / l i / + 0 , the normals to the hypersurface / = 0 are not null vectors (§ 14), and consequently the geodesics determined at each point of / = 0 by the direction of the normal are not curves of length zero. If we change coordinates taking this hypersurface for x 1 = 0 , and the geodesics for the curves of parameter x 1, and take for the coordinate x 1 the length of arc of these geodesics measured from x 1 = 0, from ( 12.5) it follows that in this coordinate system (19.1) g n == eu where ex is plus or minus one. From the equations of the geodesics which result from (17.6) when we take t = s — x 1 we have dx

0.

For i =|= 1 by hypothesis gu = 0 for x 1 = 0, it follows that gu — 0 identically. Hence the linear element is (19.2)

ds 2 = e (ei dxi + gap d x a dxP)

(a, /$ — 2, • • •, n) .

We call this the geodesic form of the linear element. As a result we have the theorem: I f f is any real function o f the x s such that A i f 4= 0 and geodesics be drawn normal to the hypersurface f — 0 and on each geodesic the same length be laid off from f — 0 , the locus o f the end points is a hypersurface orthogonal to the geodesics* These hypersurfaces are said to be geodesically parallel to the hypersurface / = 0 . Incidentally we have the theorem: * This is the generalization of a theorem of Gauss for surfaces in euclidean 3-space, cf. 1909, 1, p. 206. Also, we remark that the first assumption of the theorem is satisfied, if (12.3) is definite.

58

II. Introduction of a metric

A necessary and sufficient condition that the curves o f parameter x l be geodesic and the coordinate x l be the arc is that gn be c o n s t a n t e, and gxi for i = 2 , • • •, n be independent o f x x. For the quadratic form (19.2) we have A ix 1 — Ci.

(19.3)

Conversely, if / is any solution of the differential equation A J = elt

(19.4)

where ex is plus or minus one, the surfaces/ = const, are orthogonal to a congruence of geodesics, and the length of any geodesic between two hypersurfaces / = cu and f — c2 is c2—